In general, flow machines extract power from a fluid as the fluid flows from a higher energy state to a lower energy state. There are several types of flow machines, ranging from simple water wheels up to the most advanced gas turbines. Even though these machines have many differences, they all follow several basic laws.
Most notably, inventors from da Vinci to de Laval to Pelton have all noted that for maximum water wheel, the fluid must (1) enter slowly and evenly, or else losses will occur before the power can be extracted, (2) proceed through smooth, gradual steps, or else the power will be lost to the surroundings instead of being harvested, and (3) exit at essentially no velocity, because any residual velocity results in kinetic energy that is outside of the machine and thus cannot be converted to useful power.
Modern turbines depend on the laws of compressible flow fluid mechanics. The simplest case of these high velocity flows is the rocket.
The thrust of a rocket follows from Newtonian mechanics, where the force F is the time rate of change of momentum p, orF=dp/dt 
Because the momentum p=my, where m is the mass and v is the velocity, the force equation becomesF=dp/dt=mdv/dt+vdm/dt 
For constant velocity v, the thrust thus reduces to F=mflowv, where mflow=dm/dt is the mass flow rate.
The power P of the rocket follows from the general relation Power=Force×Velocity, or P=F·v, which becomesP=mflowv2 for the special case that the rocket velocity equals the exhaust velocity. At this condition, the exhaust hangs still in space, with no residual kinetic energy.
Using the above overall principles, the simple rocket is thus at optimum efficiency with no incoming losses, smooth expansion, and no residual kinetic energy lost in the exhaust.
Like a rocket, the thrust of a jet derives from the ejection of high speed gases. Unlike a rocket, however, a jet receives surrounding air, which then becomes most of the exhausted product; the fuel is a relatively small fraction of the exhaust gas. The incoming air velocity must therefore be subtracted from the exit velocity, leaving the power relationshipP=mflow(vexhaust−vjet)vjet where vjet is the jet velocity and vexhaust is the exhaust velocity. Note that if the incoming air were contained in the jet, the case becomes a rocket, and the jet power equation becomes the rocket power equation.
Unfortunately, applying the above general relationship of no residual energy in the exhaust requires that the exhaust velocity must equal the jet velocity. In this case, the air comes into the jet at the same velocity as the exhaust leaves the jet, and the net power is thus zero. Practical jets must therefore operate at a compromise between available thrust and efficiency.
Impulse turbines have a series of buckets mounted on a wheel. A jet of fluid directed at the buckets turns the wheel turns to generate power. Early workers found that the ideal approach is to direct the incoming stream at the edge of the bucket, not the center. The incoming fluid thus follows a semicircular path, leaving the bucket at the same speed at which the fluid entered. Furthermore, when the incoming stream enters at twice the speed of the rotating buckets, the net result is that the spent fluid is stationary in space. Thus, the maximum efficiency of an impulse turbine occurs when there is no kinetic energy left in the exhaust, as noted above for the rocket case.
Again using the relation that Power=Force×Velocity, the Power=mflowVstream×Vbucket, orP=½mflowVstream2 where mflow is the mass flow rate, Vstream is the incoming jet stream velocity, Vbucket is the bucket velocity, and Vbucket=½ Vstream. Thus, for any given jet stream velocity, an impulse turbine yields only one half of the maximum, ideal power of a rocket operating at the same velocity.
The reaction turbine is an ancient machine, originally developed by Hero of Alexandria, and most often seen today as a common rotating lawn sprinkler. The overall principle is that fluid ejected tangentially from the periphery of a disc cause rotation of this disc. Despite this inherent simplicity, the reaction turbine has never been successfully utilized for power generation. The limitations follow from the basic design. Although the original records and prototypes are long lost, the generally accepted geometry is essentially a sphere with 2 opposing jets mounted at the centerline. When water is added to the sphere, and the sphere is then heated, the escaping steam spins the machine.
The above general relationship shows the underlying problems. First, feeding in water is obviously difficult with limited machining capability. The next concern is the conversion of water to steam, and the path of this steam. Ideally, this progression should be smooth, but with a spinning mixture of liquid and gas, combined with exits that move relative to the gas (no internal partitions), the internal process is complicated indeed, leading to significant losses. Finally, there is no provision to extract the gas smoothly through and out the nozzles, and no provision to match the rotation speed of the machine to the exhaust speed. It is therefore not surprising that the Hero turbine produces no useful power. Similar problems persist in more modern attempts.
Despite these problems, reaction turbines are potentially useful because they are inherently durable. Specifically, reaction turbines lack the delicate blades that limit conventional turbines. Reaction turbines can therefore be considered for geothermal, solar, topping, bottoming, and similar cycles that have large amounts of water mixed with steam, as well as other two phase systems.
Sohre has proposed one such system, using a separator to partition the water and the steam. One limiting factor is the simple physics relationship that the linear velocity v=r ω, where r is the radius and ω is the angular velocity.
The first problem with this relationship is that Sohre specifies a supersonic tip velocity, and shows the water jet at about half the tip radius. Therefore, using the above simple equation for velocity at a given radius, the velocity at the water jet is at least near sonic, if not supersonic. This is simply not practical for multiple reasons: (1) water is not compressible, and therefore does not follow the laws of compressible flow that govern the gas phase—the gas and water components of the unit are therefore not compatible, (2) water will be subject to pump work, as described below, and (3) the amount of water in any practical two phase system is inadequate to form a useful jet—the losses would be excessive for a small diameter nozzle, but a large diameter nozzle would be inadequate to serve as a plug for the steam.
Sohre states that the ejected steam has enough radial velocity to scrub the chamber walls. One problem is that the above general relationship states that any residual velocity decreases the efficiency. Furthermore, not only does this velocity exist, it is in the radial direction. Reaction turbines, however, produce no power from radial velocity—only tangential (or angular) velocity yields useful power. The nozzle arrangement shows the reason for this loss of efficiency: the nozzles eject both tangential and radial components.
The Sohre unit does, however, show the conventional de Laval converging and diverging nozzles, as needed to reach supersonic velocities. Sohre specifies that the actual location and size depends on the pressure drop. Sohre also notes that conventional reaction turbines compress their own working fluid during rotation.
The indicated location for the proposed convergent/divergent and throat section (discussed below) is near the periphery. Specifically, the throat location is within 10% of the periphery. Therefore, the flow must progress from subsonic to sonic to supersonic in the last 10% of the rotor. The above simple relation v=r ω, however, says that such a rapid speed increase is simply not possible while maintaining the specified tangential flow equations. That is, the available space allows for at most a 10% increase in velocity, not doubling or tripling the velocity.
Proceeding farther inwards, there is a flow zone that extends from the convergent section to the axis of rotation. Sohre claims that this section may or may not have guides, depending on whether the customer wishes to pay for them. With vanes, the fluid rotates with the rotor body. Without vanes, the fluid does not rotate with the body. The fluid that approaches the nozzles therefore has entirely different flow characteristics under the two suggested configurations. Recalling the above general principle that the internal flow must be smooth to avoid excessive losses, some choice must be made here. In addition, there is also the problem of mismatched tangential velocities, which again will be discussed more fully below.
Finally, progressing towards the axis reveals one last problem that is quite instructive: mass conservation. The underlying principle here is that flow machines cannot store or release fluid. Quantitatively, the mas flow relationship ismflow=ρAv where ρ is the density, A is the cross sectional flow area, and v is the flow velocity.
The fundamental problem here is that compressible flows, such as air or steam at high velocity, have properties that limit the types of flow that are possible. Specifically, to achieve supersonic flow, as specified by Sohre, the flow must first pass through a converging section, where the velocity increases. Eventually, the flow reaches the speed of sound, Mach 1, at the narrowest point in the channel. The channel then diverges. With sufficient pressure head, the flow then accelerates beyond Mach 1 in this supersonic section. The limiting condition here is that the narrowest point is called the throat: no more than the given flow can pass through this “critical area.” Sohre's unit, however, violates this principle because the axis inlet is much smaller than the throat. Furthermore, the throat area is much, much greater than the inlet area. Therefore, Sohre's unit violates the above mass flow rate law: it is simply not physically possible for any given mass flow rate to satisfy the contradictory conditions in the Sohre geometry.
U.S. Pat. No. 5,236,349 (Fabris) presents a later approach to reaction turbines. Unfortunately, the Fabris unit begins with a problem already cited by U.S. Pat. No. 4,336,039 (Sohre): limited exhaust nozzle arrangements. Specifically, the Fabris unit has only two outlet nozzles. Most of the disc is therefore wasted space. Such an arrangement is simply not competitive on a mass and volume basis with conventional turbines.
Another problem with the Fabris unit is the inlet zone at the axis. Like the Sohre unit without partitions, this zone is not partitioned, and therefore suffers from the unequal distribution problems that date back to Hero. Furthermore, the Fabris unit also suffers from excessive efficiency losses due to high inlet velocities, as discussed more fully below.
Fabris, however, does recognize the conservation of flow limits, and the constriction to the throat is at a reasonable radial distance. Fabris also recognizes the compression effects in the rotor, primarily in regard to pressure relative to flash vaporization of the two phase (water and steam) fluid.
However, the computer program that Fabris uses to calculate these effects has flaws. First, Fabris describes a linear velocity profile, and then proceeds to use this profile for the rest of the calculations. This specification implies that either the linear profile is the only possible profile, or there is some preference for the linear profile. Actually, there are many possible profiles, and many of them provide significant improvements over the linear form specified by Fabris (discussed more fully below).
More importantly, Fabris then uses this linear profile in an iterative scheme to determine the flow contours. The iteration limit is the predicted gas velocity in the moving rotor, versus the gas velocity in a stationary rotor. Fabris then uses the converged case for testing.
The difficulty here is that Fabris has omitted crucial terms in the computational model: losses and the work done by the gas on the rotor in the tangential direction prior to the exit. Because these terms decrease the exit velocity in the rotating case, the Fabris algorithm of mandated identical exit velocities invariably converges to the wrong solution. This topic is discussed more fully below, but for here, note that a 180 degree change of gas direction in the tangential rotation plane (Fabris FIG. 3) yields results that are in error by at least 50%. Such large discrepancies are simply unacceptable in turbine work, where variations of even fractions of a percent are important.
Beyond the patent literature, Comfort provides the simplest, most complete analysis of the principles of the reaction turbine. In particular, Comfort cites the omission of the pump work in most text books.